Linear.Quaternion:$cexp from linear-1.19.1.3

Average Error: 0.1 → 0.2

Time: 1.7min

Precision: binary64

Cost: 13252

Math

\[x \cdot \frac{\sin y}{y} \]

↓

\[\begin{array}{l} \mathbf{if}\;\sin y \ne 0:\\ \;\;\;\;\frac{x}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \sin y\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

↓

(FPCore (x y)
 :precision binary64
 (if (!= (sin y) 0.0) (/ x (/ y (sin y))) (* (/ x y) (sin y))))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

↓

double code(double x, double y) {
	double tmp;
	if (sin(y) != 0.0) {
		tmp = x / (y / sin(y));
	} else {
		tmp = (x / y) * sin(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (sin(y) / y)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sin(y) /= 0.0d0) then
        tmp = x / (y / sin(y))
    else
        tmp = (x / y) * sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return x * (Math.sin(y) / y);
}

↓

public static double code(double x, double y) {
	double tmp;
	if (Math.sin(y) != 0.0) {
		tmp = x / (y / Math.sin(y));
	} else {
		tmp = (x / y) * Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y):
	return x * (math.sin(y) / y)

↓

def code(x, y):
	tmp = 0
	if math.sin(y) != 0.0:
		tmp = x / (y / math.sin(y))
	else:
		tmp = (x / y) * math.sin(y)
	return tmp

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

↓

function code(x, y)
	tmp = 0.0
	if (sin(y) != 0.0)
		tmp = Float64(x / Float64(y / sin(y)));
	else
		tmp = Float64(Float64(x / y) * sin(y));
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = x * (sin(y) / y);
end

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sin(y) ~= 0.0)
		tmp = x / (y / sin(y));
	else
		tmp = (x / y) * sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := If[Unequal[N[Sin[y], $MachinePrecision], 0.0], N[(x / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.1

   \[x \cdot \frac{\sin y}{y} \]

2. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\sin y \ne 0:\\ \;\;\;\;\frac{x}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{y}\\ } \end{array}} \]

3. Simplified 0.2

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\sin y \ne 0:\\ \;\;\;\;\frac{x}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \sin y\\ } \end{array}} \]
   Proof

Alternatives

Alternative 1
Error	0.1
Cost	6720

\[x \cdot \frac{\sin y}{y} \]

Alternative 2
Error	23.8
Cost	716

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{y}\\ \end{array}\\ \mathbf{if}\;y \leq -10000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{+42}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	23.8
Cost	716

\[\begin{array}{l} t_0 := \frac{y \cdot x}{y}\\ \mathbf{if}\;y \leq -22:\\ \;\;\;\;3 - \left(3 - t_0\right)\\ \mathbf{elif}\;y \leq 10^{-31}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	24.4
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{y} \cdot y\\ \mathbf{if}\;y \leq -2 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{+42}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	30.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))